Extending Pauling's estimate [1]} can become extremely challenging on arbitrary graphs. In order to estimate the degeneracy of the ice manifold beyond the case of planar graphs or non-bipartite graphs, we will use a general graph-theoretic approach, and upper bound the entropy associated to the ice manifold. First, let us note that the maximum degeneracy that a spin ice can have is given by \(2^{N_{spins}}\) . If the spin ice is degree regular and has \(N_v\)  vertices, then the maximum entropy of the ice manifold is naturally given by \(\epsilon _{max}=N_v \frac{d_v}{2} \ln 2. \) 
Below we provide a procedure to systematically calculate upper bounds based on the theory of Eulerian tours on graphs. Given a certain spin ice graph \(\mathcal {G}\) , we are interested in calculating the number of configurations in the ice manifold, \(\epsilon (\mathcal {G})\) , and its entropy \(S(\mathcal {G})=\ln \epsilon (\mathcal {G})\)  [1]}.
